After many heads in a row, is tails more likely on the next flip?
Short answer: Independent trials have no memory. Every flip is ½.
The trap
The wrong idea
“After many heads, tails becomes more likely.”
It sounds reasonable, and a lot of JEE aspirants carry it into the exam. Here's a one-question check, then the explanation that makes it click.
Quick check
Try it before you peek
A fair coin shows heads 5 times in a row. P(tails next) is:
- More than ½
- Exactly ½
- Less than ½
Show the answer
B. Exactly ½
Independent trials have no memory. Every flip is ½.
The click
For a fair coin, each flip is an independent event: the coin has no memory of earlier results. The probability of tails on the next flip is ½, whether the previous five flips were all heads, all tails or a mix. Believing otherwise is known as the gambler's fallacy.
A quick calculation shows it. Before any flips, the probability of five heads followed by a tail, HHHHHT, is (½)⁶ = 1/64. The probability of six heads, HHHHHH, is also (½)⁶ = 1/64. Once the first five heads have happened, these are the only two possibilities left, and they are equally likely, so P(tails next) = (1/64) ÷ (2/64) = ½.
Why it feels wrong: a run of six heads is rare (1 in 64) when you judge it before any flips. But once five heads have already happened, that rarity is used up; only the sixth flip is still uncertain.
The law of large numbers says the proportion of heads approaches ½ over many flips. It does this by dilution, not compensation: later flips don't balance out a streak, they simply swamp it as the total number of flips grows. If a long streak makes you suspect the coin isn't fair, that is a different question about the coin itself, and it would point toward more heads, not toward tails being due.
Got this one? The Knot Scan finds the misconception you don't know you have, in 8 questions.